Factorization Comparison Theorems and Their Limits

Factorization and algebraic QFT can be compared only after their geometries, target categories, locality operations, and weak equivalences are aligned. There is a genuine one-categorical equivalence theorem for additive, Cauchy-constant AQFTs and additive, Cauchy-constant time-orderable prefactorization algebras. There are also explicit cochain comparisons for free Green-hyperbolic theories and perturbative observables. None of these statements is a framework-wide equivalence between arbitrary Euclidean factorization algebras and arbitrary operator-algebraic nets.

Required background. Typed comparisons of LCQFT, nets, fields, and factorization fixes the categories being compared; prefactorization and factorization algebras supplies disjoint products; and Weiss descent distinguishes full factorization from weaker additivity.

Helpful background. Equivalence and comparison notions separates isomorphism, quasi-isomorphism, and categorical equivalence, while rigorous RG scheme comparison gives a different kind of continuum comparison.

Aligning the comparison data

An AQFT on the category $\mathbf{Loc}$ of oriented, time-oriented globally hyperbolic spacetimes is a covariant functor $\mathfrak A:\mathbf{Loc}\to\mathbf{Alg}(\mathcal C)$ satisfying Einstein causality. A time-orderable prefactorization algebra $\mathfrak F$ assigns objects of $\mathcal C$ and products only to disjoint tuples admitting a causal ordering. To compare them, one must state at least:

• whether $\mathcal C$ is vector spaces, cochain complexes, nuclear locally convex complexes, or operator algebras;
• whether Cauchy morphisms act by isomorphisms or merely quasi-isomorphisms;
• which relatively compact causally convex opens generate the global value;
• whether the prefactorization object has ordinary Weiss descent, Lorentzian additivity, or both; and
• whether an involution, positivity, completion, and states are part of the target.

Erasing any row changes the theorem. In particular, a quasi-isomorphism of observable complexes is not an isomorphism of completed $C^*$-algebras, and neither statement identifies state spaces automatically.

The model-independent strict theorem

Write $\mathbf{AQFT}^{\mathrm{add},c}$ for additive Cauchy-constant AQFTs and $\mathbf{tPFA}^{\mathrm{add},c}$ for additive Cauchy-constant time-orderable prefactorization algebras in a suitable bicomplete closed symmetric monoidal one-category. There are functors

$$\mathbb F:\mathbf{AQFT}^{\mathrm{add},c} \rightleftarrows \mathbf{tPFA}^{\mathrm{add},c}:\mathbb A.$$

$\mathbb F$ uses multiplication in the AQFT, ordered according to causal position. $\mathbb A$ uses the time-slice property to construct an associative multiplication from a prefactorization product. Additivity makes this multiplication natural and proves Einstein causality. The two functors are inverse in the strict setting Benini, Perin, and Schenkel 2020, Theorems 3.11, 4.7, and 5.1. An involution can be transferred on this restricted subcategory, but its definition explicitly uses Cauchy constancy.

The word “time-orderable” is decisive. The theorem does not identify an arbitrary Costello–Gwilliam factorization algebra on a Riemannian manifold with an AQFT. Nor does additive prefactorization automatically imply full Weiss descent; every factorization algebra in the cited setting is additive, while the converse requires more.

Free and perturbative comparisons

For a free scalar or another Green-hyperbolic linear theory on a fixed globally hyperbolic spacetime, the pAQFT net and the factorization construction admit natural transformations that are isomorphisms of the underlying cochain complexes. Time-ordered multiplication supplies the bridge, and the time-slice property recovers the associative algebra on a Cauchy surface Gwilliam and Rejzner 2020, Theorems 3.5–3.6 and 3.17. This comparison applies before taking a Hilbert-space representation.

For perturbatively interacting theories, renormalized multilocal functionals and the anomalous master Ward identity produce a factorization algebra Gwilliam and Rejzner 2023, §§4–6. The equality is again formal in the coupling and $\hbar$ unless a separate convergence result is supplied.

The free-scalar application belongs at Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks. For that model, one can compare microcausal cochains, time-ordered products, and net observables. Agreement on cohomology does not imply equality of off-shell complexes or identify every completion.

Adversarial extension test

Take a nonperturbative type-III Haag–Kastler net and ask the strict theorem to produce a Euclidean factorization algebra with BV differential, Weiss descent, and a path-integral state. The input supplies none of the required cochain resolution, Euclidean continuation, or factorization products on non-time-orderable configurations. The theorem can produce only the associated time-orderable prefactorization structure in its stated target, provided additivity and Cauchy constancy hold. Requiring more exposes a missing construction, not a weakness in the net.

Conversely, a locally constant $E_n$ factorization algebra can have perfect descent while lacking a $*$-operation, positivity, causal geometry, and time-slice maps. It therefore does not become a physical AQFT solely by changing vocabulary.

Exercises

Why does Weiss descent imply the additivity used in the strict comparison under the cited hypotheses?

Solution

Relatively compact causally convex opens form a Weiss-type generating family for finite supports. Descent makes the global value the colimit of the corresponding local diagram, which is the additivity condition. The reverse implication need not reconstruct higher Čech relations.

Name the weakest conclusion of the free-field natural transformation.

Solution

It identifies the specified underlying cochain complexes naturally and intertwines the stated products after the time-ordering comparison. It does not by itself identify Hilbert representations, states, or $C^*$-completions.

References

• Benini, Marco, Marco Perin, and Alexander Schenkel. “Model-Independent Comparison between Factorization Algebras and Algebraic Quantum Field Theory on Lorentzian Manifolds.” Communications in Mathematical Physics 377 (2020): 971–997. arXiv:1903.03396.
• Gwilliam, Owen, and Katarzyna Rejzner. “Relating Nets and Factorization Algebras of Observables: Free Field Theories.” Communications in Mathematical Physics 373 (2020): 107–174. doi:10.1007/s00220-019-03652-9.
• Gwilliam, Owen, and Katarzyna Rejzner. “The Observables of a Perturbative Algebraic Quantum Field Theory Form a Factorization Algebra.” 2023. arXiv:2212.08175.